Taylor and Maclaurin Series 1 Taylor and Maclaurin Series If we represent some function f(x) as a power series in (x-a), then 2 Uniqueness Theorem Suppose for every x in some interval around a. Then . 3 Taylor's Formula with Remainder Let f(x) be a function such that f(n+1)(x) exists for all x on an open interval containing a. Then, for every x in the interval, where Rn(x) is the remainder (or error). Taylor's Theorem Let f be a function with all derivatives in (a-r,a+r). The Taylor Series represents f(x) on (a-r,a+r) if and only if . 4 EX 1 Find the Maclaurin series for f(x)=cos x and prove it represents cos x for all x. 5 EX 2 Find the Maclaurin series for f(x) = sin x. 6 EX 3 Write the Taylor series for centered at a=1. 7 EX 4 Find the Taylor series for f(x) = sin x in (x-π/4). 8 EX 5 Use what we already know to write a Maclaurin series (5 terms) for . 9 10 11